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The '''Hitchin functional''' is a [[mathematics|mathematical]] concept with applications in [[string theory]] that was introduced by the British [[mathematician]] [[Nigel Hitchin]]. {{harvtxt|Hitchin|2000}} and {{harvtxt|Hitchin|2001}} are the original articles of the Hitchin functional.
 
As with Hitchin's introduction of [[generalized complex manifold]]s, this is an example of a mathematical tool found useful in [[mathematical physics]].
 
== Formal definition ==
This is the definition for 6-manifolds. The definition in Hitchin's article is more general, but more abstract.<ref>For explicitness, the definition of ''Hitchin functional is written before some explanations.</ref>
 
Let <math>M</math> be a [[compact space|compact]], [[oriented]] 6-[[manifold]] with trivial [[canonical bundle]]. Then the '''Hitchin functional''' is a [[functional (mathematics)|functional]] on [[differential form|3-forms]] defined by the formula:
 
: <math>\Phi(\Omega) = \int_M \Omega \wedge * \Omega,</math>
 
where <math>\Omega</math> is a 3-form and * denotes the [[Hodge star]] operator.
 
== Properties ==
* The Hitchin functional is analogous for six-manifold to the [[Yang-Mills]] functional for the four-manifolds.
 
* The Hitchin functional is manifestly [[invariant (mathematics)|invariant]] under the [[group action|action]] of the [[group (mathematics)|group]] of orientation-preserving [[diffeomorphism]]s.
 
* '''Theorem.''' Suppose that <math>M</math> is a three-dimensional [[complex manifold]] and <math>\Omega</math> is the real part of a non-vanishing [[holomorphic]] 3-form, then <math>\Omega</math> is a [[critical point (mathematics)|critical point]] of the functional <math>\Phi</math> restricted to the [[cohomology class]] <math>[\Omega] \in H^3(M,R)</math>. Conversely, if <math>\Omega</math> is a critical point of the functional <math>\Phi</math> in a given comohology class and <math>\Omega \wedge * \Omega < 0</math>, then <math>\Omega</math> '''defines''' the structure of a complex manifold, such that <math>\Omega</math> is the real part of a non-vanishing holomorphic 3-form on <math>M</math>.
 
:The proof of the theorem in Hitchin's articles {{harvs|txt|last=Hitchin|year=2000}} and {{harvs|txt|last=Hitchin|year=2001}} is relatively straightforward. The power of this concept is in the converse statement: if the exact form <math>\Phi(\Omega)</math> is known, we only have to look at its critical points to find the possible complex structures.
 
== Stable forms ==
Action functionals often determine geometric structure<ref>For example, complex structure, symplectic structure, <math>G_2</math> holonomy and <math>Spin(7)</math> holonomy etc.</ref> on <math>M</math> and geometric structure are often characterized by the existence of particular differential forms on <math>M</math> that obey some integrable conditions.
 
If an ''m''-form <math>\omega</math> can be written with local coordinates
:<math>\omega=dp_1\wedge dq_1+\cdots+dp_m\wedge dq_m</math>
and
:<math>d\omega=0</math>,
then <math>\omega</math> defines ''[[symplectic manifold|symplectic structure]]''.
 
A ''p''-form <math>\omega\in\Omega^p(M,\mathbb{R})</math> is ''stable'' if it lies in an open orbit of the local <math>GL(n,\mathbb{R})</math> action where n=dim(M), namely if any small perturbation <math>\omega\mapsto\omega+\delta\omega</math> can be undone by a local <math>GL(n,\mathbb{R})</math> action. So any ''1''-form that don't vanish everywhere is stable; ''2''-form (or ''p''-form when ''p'' is even) stability is equivalent to nondegeneratacy.
 
What about ''p''=3? For large ''n'' ''3''-form is difficult because the dimension of <math>\wedge^3(\mathbb{R}^n)</math>, <math>n^3</math>, grows more firstly than the dimension of <math>GL(n,\mathbb{R})</math>, <math>n^2</math>. But there are some very lucky exceptional case, namely, <math>n=6</math>, when dim <math>\wedge^3(\mathbb{R}^6)=20</math>, dim <math>GL(6,\mathbb{R})=36</math>. Let <math>\rho</math> be a stable real ''3''-form in dimension ''6''. Then the stabilizer of <math>\rho</math> under <math>GL(6,\mathbb{R})</math> has real dimension ''36-20=16'', in fact either <math>SL(3,\mathbb{R})\times SL(3,\mathbb{R})</math> or <math>SL(3,\mathbb{C})\cap SL(3,\mathbb{C})</math>.
 
Focus on the case of <math>SL(3,\mathbb{C})\cap SL(3,\mathbb{C})</math> and if <math>\rho</math> has a stabilizer in <math>SL(3,\mathbb{C})\cap SL(3,\mathbb{C})</math> then it can be written with local coordinates as follows:
:<math>\rho=\frac{1}{2}(\zeta_1\wedge\zeta_2\wedge\zeta_3+\bar{\zeta_1}\wedge\bar{\zeta_2}\wedge\bar{\zeta_3})</math>
where <math>\zeta_1=e_1+ie_2,\zeta_2=e_3+ie_4,\zeta_3=e_5+ie_6</math> and <math>e_i</math> are bases of <math>T^*M</math>. Then <math>\zeta_i</math> determines an [[almost complex structure]] on <math>M</math>. Moreover, if there exist local coordinate <math>(z_1,z_2,z_3)</math> such that <math>\zeta_i=dz_i</math> then it determines fortunately an [[complex manifold|complex structure]] on <math>M</math>.
 
Given the stable <math>\rho\in\Omega^3(M,\mathbb{R})</math>:
:<math>\rho=\frac{1}{2}(\zeta_1\wedge\zeta_2\wedge\zeta_3+\bar{\zeta_1}\wedge\bar{\zeta_2}\wedge\bar{\zeta_3})</math>.
We can define another real ''3''-from
:<math>\tilde{\rho}(\rho)=\frac{1}{2}(\zeta_1\wedge\zeta_2\wedge\zeta_3-\bar{\zeta_1}\wedge\bar{\zeta_2}\wedge\bar{\zeta_3})</math>.
 
And then <math>\Omega=\rho+i\tilde{\rho}(\rho)</math> is a holomorphic ''3''-form in the almost complex structure determined by <math>\rho</math>. Furthermore, it becomes to be the complex structure just if <math>d\Omega=0</math> i.e. <math>d\rho=0</math>
and <math>d\tilde{\rho}(\rho)=0</math>. This <math>\Omega</math> is just the ''3''-form <math>\Omega</math> in formal definition of ''Hitchin functional''. These idea induces the [[generalized complex structure]].
 
== Use in string theory ==
Hitchin functionals arise in many areas of string theory. An example is the [[compactification (physics)|compactifications]] of the 10-dimensional string with a subsequent [[orientifold]] projection <math>\kappa</math> using an [[Involution (mathematics)|involution]] <math>\nu</math>. In this case, <math>M</math> is the internal 6 (real) dimensional [[Calabi-Yau space]]. The couplings to the complexified [[Kähler manifold|Kähler coordinates]] <math>\tau</math> is given by
 
: <math>g_{ij} = \tau \text{im} \int \tau i^*(\nu \cdot \kappa \tau).</math>
 
The potential function is the functional <math>V[J] = \int J \wedge J \wedge J</math>, where J is the [[almost complex structure]]. Both are Hitchin functionals.{{harvtxt|Grimm|Louis|2004}}
 
As application to string theory, the famous OSV conjecture {{harvtxt|Ooguri|Strominger|Vafa|2004}} used ''Hitchin functional'' in order to relate topological string to 4-dimensional black hole entropy. Using similar technique in the <math>G_2</math> holonomy {{harvtxt|Dijkgraaf|Gukov|Neitake|Vafa|2004}} argued about [[Topological string theory#Topological M-theory|topological M-theory]] and in the <math>Spin(7)</math> holonomy topological F-theory might be argued.
 
More recently, [[E. Witten]] claimed the mysterious superconformal field theory in six dimensions.{{harvtxt|Witten|2007}}
 
== Notes ==
<references/>
 
== References ==
*{{citation|title=The geometry of three-forms in six and seven dimensions
|journal=
|publisher=
|volume=| issue =|year= 2000
|doi=
|pages=
|url=http://arxiv.org/abs/math/0010054
|authorlink=Nigel Hitchin
|first=Nigel |last=Hitchin}}
*{{citation|title=Stable forms and special metric
|journal=
|publisher=
|volume=| issue =|year= 2001
|doi=
|pages=
|url=http://arxiv.org/abs/math/0107101
|authorlink=Nigel Hitchin
|first=Nigel |last=Hitchin}}
*{{citation|title=The effective action of Type IIA Calabi-Yau orientifolds
|author=Thomas Grimm; Jan Louis
|journal=Nuclear Physics B
|publisher=Institut für Theoretische Physik
|volume=718 | issue=1–2 |year= 2005
|doi=10.1016/j.nuclphysb.2005.04.007
|pages=153–202 |url=http://arxiv.org/abs/hep-th/0412277}}
*{{citation|title=Topological M-theory as Unification of Form Theories of Gravity
|author=Robert Dijikgraaf; Sergei Gukov; Andrew Neitzke & Cumrun Vafa
|journal=
|publisher=
|volume=| issue =|year= 2004
|doi=
|pages=
|url=http://arxiv.org/pdf/hep-th/0411073}}
*{{citation|title=Black Hole Attractors and the Topological String
|journal=
|publisher=
|volume=| issue =|year= 2004
|doi=
|pages=
|url=http://arxiv.org/abs/hep-th/0405146
|author=Hiroshi Ooguri, Andrew Strominger & Cumran Vafa }}
*{{citation|title=Conformal Field Thoery In Four And Six Dimensions
|first=Edward |last=Witten
|journal=
|publisher=
|volume=| issue =|year= 2007
|doi=
|pages=
|url=http://arxiv.org/abs/0712.0157v2
}}
 
{{DEFAULTSORT:Hitchin Functional}}
[[Category:Complex manifolds]]
[[Category:Symplectic manifolds]]
[[Category:String theory]]

Revision as of 17:19, 9 January 2014

The Hitchin functional is a mathematical concept with applications in string theory that was introduced by the British mathematician Nigel Hitchin. Template:Harvtxt and Template:Harvtxt are the original articles of the Hitchin functional.

As with Hitchin's introduction of generalized complex manifolds, this is an example of a mathematical tool found useful in mathematical physics.

Formal definition

This is the definition for 6-manifolds. The definition in Hitchin's article is more general, but more abstract.[1]

Let be a compact, oriented 6-manifold with trivial canonical bundle. Then the Hitchin functional is a functional on 3-forms defined by the formula:

where is a 3-form and * denotes the Hodge star operator.

Properties

  • The Hitchin functional is analogous for six-manifold to the Yang-Mills functional for the four-manifolds.
The proof of the theorem in Hitchin's articles Template:Harvs and Template:Harvs is relatively straightforward. The power of this concept is in the converse statement: if the exact form is known, we only have to look at its critical points to find the possible complex structures.

Stable forms

Action functionals often determine geometric structure[2] on and geometric structure are often characterized by the existence of particular differential forms on that obey some integrable conditions.

If an m-form can be written with local coordinates

and

,

then defines symplectic structure.

A p-form is stable if it lies in an open orbit of the local action where n=dim(M), namely if any small perturbation can be undone by a local action. So any 1-form that don't vanish everywhere is stable; 2-form (or p-form when p is even) stability is equivalent to nondegeneratacy.

What about p=3? For large n 3-form is difficult because the dimension of , , grows more firstly than the dimension of , . But there are some very lucky exceptional case, namely, , when dim , dim . Let be a stable real 3-form in dimension 6. Then the stabilizer of under has real dimension 36-20=16, in fact either or .

Focus on the case of and if has a stabilizer in then it can be written with local coordinates as follows:

where and are bases of . Then determines an almost complex structure on . Moreover, if there exist local coordinate such that then it determines fortunately an complex structure on .

Given the stable :

.

We can define another real 3-from

.

And then is a holomorphic 3-form in the almost complex structure determined by . Furthermore, it becomes to be the complex structure just if i.e. and . This is just the 3-form in formal definition of Hitchin functional. These idea induces the generalized complex structure.

Use in string theory

Hitchin functionals arise in many areas of string theory. An example is the compactifications of the 10-dimensional string with a subsequent orientifold projection using an involution . In this case, is the internal 6 (real) dimensional Calabi-Yau space. The couplings to the complexified Kähler coordinates is given by

The potential function is the functional , where J is the almost complex structure. Both are Hitchin functionals.Template:Harvtxt

As application to string theory, the famous OSV conjecture Template:Harvtxt used Hitchin functional in order to relate topological string to 4-dimensional black hole entropy. Using similar technique in the holonomy Template:Harvtxt argued about topological M-theory and in the holonomy topological F-theory might be argued.

More recently, E. Witten claimed the mysterious superconformal field theory in six dimensions.Template:Harvtxt

Notes

  1. For explicitness, the definition of Hitchin functional is written before some explanations.
  2. For example, complex structure, symplectic structure, holonomy and holonomy etc.

References

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    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010